Example 108.20.1. Let $k$ be a field, and let $A = k[x_1, x_2, x_3, \dots ] / (x_1^2, x_2^2, x_3^2, \dots )$. Any prime ideal of $A$ contains the nilpotents $x_1, x_2, x_3, \dots $, so $\mathfrak p = (x_1, x_2, x_3, \dots )$ is the only prime ideal of $A$. Therefore the underlying topological space of $\operatorname {Spec} A$ is a single point and in particular is Noetherian. However $\mathfrak p$ is clearly not finitely generated.

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